10
GEOMETRIC GRAPH THEORY
János Pach
INTRODUCTION
In the traditional areas of graph theory (Ramsey theory, extremal graph theory,
random graphs, etc.), graphs are regarded as abstract binary relations. The relevant
methods are often incapable of providing satisfactory answers to questions arising
in geometric applications. Geometric graph theory focuses on combinatorial and
geometric properties of graphs drawn in the plane by straight-line edges (or, more
generally, by edges represented by simple Jordan arcs). It is a fairly new discipline
abounding in open problems, but it has already yielded some striking results that
have proved instrumental in the solution of several basic problems in combinatorial
and computational geometry (including the k-set problem and metric questions
discussed in Sections 1.1 and 1.2, respectively, of this Handbook). This chapter is
partitioned into extremal problems (Section 10.1), crossing numbers (Section 10.2),
and generalizations (Section 10.3).
10.1 EXTREMAL PROBLEMS
Turán’s classical theorem [Tur54] determines the maximum number of edges that
an abstract graph with n vertices can have without containing, as a subgraph, a
complete graph with k vertices. In the spirit of this result, one can raise the following general question. Given a class H of so-called forbidden geometric subgraphs,
what is the maximum number of edges that a geometric graph of n vertices can have
without containing a geometric subgraph belonging to H? Similarly, Ramsey’s theorem [Ram30] for abstract graphs has some natural analogues for geometric graphs.
In this section we will be concerned mainly with problems of these two types.
GLOSSARY
Geometric graph: A graph drawn in the plane by (possibly crossing) straightline segments; i.e., a pair (V (G), E(G)), where V (G) is a set of points (‘vertices’),
no three of which are collinear, and E(G) is a set of segments (‘edges’) whose
endpoints belong to V (G).
Convex geometric graph: A geometric graph whose vertices are in convex position; i.e., they form the vertex set of a convex polygon.
Cyclic chromatic number of a convex geometric graph: The minimum number
χc (G) of colors needed to color all vertices of G so that each color class consists
of consecutive vertices along the boundary of the convex hull of the vertex set.
Convex matching: A convex geometric graph consisting of disjoint edges, each
of which belongs to the boundary of the convex hull of its vertex set.
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J. Pach
Parallel matching: A convex geometric graph consisting of disjoint edges, the
convex hull of whose vertex set contains only two of the vertices on its boundary.
Complete geometric
graph: A geometric graph G whose edge set consists of
all |V (G)|
segments
between
its vertices.
2
Complete bipartite geometric graph: A geometric graph G with V (G) =
V1 ∪ V2 , whose edge set consists of all segments between V1 and V2 .
Geometric subgraph of G:
E(H) ⊆ E(G).
A geometric graph H, for which V (H) ⊆ V (G) and
Crossing: A common interior point of two edges of a geometric graph.
(k, l)-Grid: k + l vertex-disjoint edges in a geometric graph such that each of
the first k edges crosses all of the last l edges. It is called natural if the first
k edges are pairwise disjoint segments and the last l edges are pairwise disjoint
segments.
Disjoint edges: Edges of a geometric graph that do not cross and do not even
share an endpoint.
Parallel edges: Edges of a geometric graph whose supporting lines are parallel or
intersect at points not belonging to any of the edges (including their endpoints).
x-Monotone curve: A continuous curve that intersects every vertical line in at
most one point.
Outerplanar graph: A (planar) graph that can be drawn in the plane without crossing so that all points representing its vertices lie on the outer face of
the resulting subdivision of the plane. A maximal outerplanar graph is a
triangulated cycle.
Hamiltonian path: A path going through all elements of a finite set S. If the
elements of S are colored by two colors, and no two adjacent elements of the
path have the same color, then it is called an alternating path.
Hamiltonian cycle: A cycle going through all elements of a finite set S.
Caterpillar: A tree consisting of a path P and of some extra edges, each of which
is adjacent to a vertex of P .
CROSSING-FREE GEOMETRIC GRAPHS
1. Hanani-Tutte theorem: Any graph that can be drawn in the plane so that its
edges are represented by simple Jordan arcs such that any two that do not
share an endpoint properly cross an even number of times, is planar [Cho34,
Tut70]. The analogous result also holds in the projective plane [PSS09].
2. Fáry’s theorem: Every planar graph admits a crossing-free straight-line drawing [Fár48, Tut60, Ste22]. Moreover, every 3-connected planar graph and its
dual have simultaneous straight-line drawings in the plane such that only dual
pairs of edges cross and every such pair is perpendicular [BS93].
3. Koebe’s theorem: The vertices of every planar graph can be represented
by nonoverlapping disks in the plane such that two of them are tangent
to each other if and only if the corresponding two vertices are adjacent
[Koe36, Thu78]. This immediately implies Fáry’s theorem.
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Chapter 10: Geometric graph theory
259
4. Pach-Tóth theorem: Any graph that can be drawn in the plane so that its
edges are represented by x-monotone curves with the property that any two
of them either share an endpoint or properly cross an even number of times
admits a crossing-free straight-line drawing, in which the x-coordinates of the
vertices remain the same [PT04]. Fulek et al. [FPSŠ13] generalized this result
in two directions: it is sufficient to assume that (a) any two edges that do not
share an endpoint cross an even number of times, and that (b) the projection
of every edge to the x-axis lies between the projections of its endpoints.
5. Grid drawings of planar graphs: Every planar graph of n vertices admits a
straight-line drawing such that the vertices are represented by points belonging to an (n−1) × (n−1) grid [FPP90, Sch90]. Furthermore, such a drawing
can be found in O(n) time. For other small-area grid drawings, consult [?].
6. Straight-line drawings of outerplanar graphs: For any outerplanar graph H
with n vertices and for any set P of n points in the plane in general position,
there is a crossing-free geometric graph G with V (G) = P , whose underlying
graph is isomorphic to H [GMPP91]. For any rooted tree T and for any set
P of |V (T )| points in the plane in general position with a specified element
p ∈ P , there is a crossing-free straight-line drawing of T such that every
vertex of T is represented by an element of P and the root is represented by
p [IPTT94]. This theorem generalizes to any pair of rooted trees, T1 and T2 :
for any set P of n = |V (T1 )| + |V (T2 )| points in general position in the plane,
there is a crossing-free mapping of T1 ∪ T2 that takes the roots to arbitrarily
prespecified elements of P . Such a mapping can be found in O(n2 log n) time
[KK00]. The analogous statement for triples of trees is false.
7. Alternating paths: Given n red points and n blue points in general position
in the plane, separated by a straight line, they always admit a noncrossing
alternating Hamiltonian path [KK03].
TURÁN-TYPE PROBLEMS
By Euler’s Polyhedral Formula, if a geometric graph G with n ≥ 3 vertices has
no 2 crossing edges, it cannot have more than 3n − 6 edges. It was shown in
[AAP+ 97] and [Ack09] that under the weaker condition that no 3 (resp. 4) edges
are pairwise crossing, the number of edges of G is still O(n). It is not known whether
this statement remains true even if we assume only that no 5 edges are pairwise
crossing. As for the analogous problem when the forbidden configuration consists of
k pairwise disjoint edges, the answer is linear for every k [PT94]. In particular, for
k = 2, the number of edges of G cannot exceed the number of vertices [HP34]. The
best lower and upper bounds known for the number of edges of a geometric graph
with n vertices, containing no forbidden geometric subgraph of a certain type, are
summarized in Table 10.1.1. The letter k always stands for a fixed positive integer
parameter and n tends to infinity. Wherever k does not appear in the asymptotic
bounds, it is hidden in the constants involved in the O- and Ω-notations.
Better results are known for convex geometric graphs, i.e., when the vertices are
in convex position. The relevant bounds are listed in Table 10.1.2. For any convex
geometric graph G, let χc (G) denote its cyclic chromatic number. Furthermore, let
ex(n, Kk ) stand for the maximum number of edges of a graph with n vertices that
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J. Pach
TABLE 10.1.1
Maximum number of edges of a geometric graph of n vertices containing
no forbidden subconfigurations of a certain type.
FORBIDDEN CONFIGURATION
2 crossing edges
3 pairwise crossing edges
4 pairwise crossing edges
k > 4 pairwise crossing edges
an edge crossing 2 others
an edge crossing 3 others
an edge crossing 4 others
an edge crossing 5 others
an edge crossing k others
2 crossing edges crossing k others
(k, l)-grid
natural (k, l)-grid
self-intersecting path of length 3
self-intersecting path of length 5
self-intersecting cycle of length 4
2 disjoint edges
noncrossing path of length k
k pairwise parallel edges
LOWER BOUND
UPPER BOUND
SOURCE
3n − 6
6.5n − O(1)
Ω(n)
Ω(n)
4n − 8
5n − 12
5.5n + Ω(1)
6n −
√O(1)
Ω( kn)
Ω(n)
Ω(n)
Ω(n)
Ω(n log n)
Ω(n log log n)
Ω(n3/2 )
n
Ω(kn)
Ω(n)
3n − 6
6.5n − O(1)
72(n − 2)
O(n log n)
4n − 8
5n − 10
5.5n + O(1)
6n√
− 12
O( kn)
O(n)
O(n)
O(n log2 n)
O(n log n)
O(n log n/ log log n)
O(n3/2 log n)
n
O(k 2 n)
O(n)
Euler
[AT07]
[Ack09]
[Val98]
[PT97]
[PT97]
[PRTT06]
[Ack16]
[PT97]
[PRT04]
[PPST05]
[AFP+ 14]
[PPTT02]
[Tar13], [PPTT02]
[PR04], [MT06]
[HP34]
[Tót00]
[Val98]
FIGURE 10.1.1
Geometric graph with n = 20 vertices and 5n − 12 = 88 edges,
none of which crosses 3 others.
does not have a complete subgraph with
k vertices. By Turán’s theorem [Tur54]
n
+
O(n)
is equal to the number of edges of
mentioned above, ex(n, Kk ) = k−2
k−1 2
a complete (k−1)-partite graph with n vertices whose vertex classes are of size
⌊n/(k − 1)⌋ or ⌈n/(k − 1)⌉. Two disjoint self-intersecting paths of length 3, xyvz
and x′ y ′ v ′ z ′ , in a convex geometric graph are said to be of the same orientation
if the cyclic order of their vertices is x, v, x′ , v ′ , y ′ , z ′ , y, z (
). They are said to
have opposite orientations if the cyclic order of their vertices is x, v, v ′ , x′ , z ′ , y ′ , y, z
(type 1:
) or v, x, x′ , v ′ , y ′ , z ′ , z, y (type 2:
).
RAMSEY-TYPE PROBLEMS
In classical Ramsey theory, one wants to find large monochromatic subgraphs in a
complete graph whose edges are colored with several colors [GRS90]. Most questions
Preliminary version (December 12, 2016).
Chapter 10: Geometric graph theory
TABLE 10.1.2
261
Maximum number of edges of a convex geometric graph of n vertices containing no
forbidden subconfigurations of a certain type.
FORBIDDEN CONFIGURATION
2 crossing edges
self-intersecting path of length 3
k self-intersecting paths of length 3
2 self-intersecting paths of length 3
with opposite orientations of type 1
2 self-intersecting paths of length 3
with opposite orientations of type 2
2 adjacent edges crossing a 3rd
k pairwise crossing edges
noncrossing outerplanar graph of k
vertices, having a Hamiltonian cycle
convex geometric subgraph G
convex matching of k disjoint edges
parallel matching of k disjoint edges
noncrossing caterpillar C of k vertices
LOWER BOUND
UPPER BOUND
SOURCE
2n − 3
2n − 3
Ω(n)
Ω(n log n)
2n − 3
2n − 3
O(n)
O(n log n)
Euler
Perles
[BKV03]
[BKV03]
Ω(n log n)
O(n log n)
[BKV03]
⌊5n/2 − 4⌋
2(k−1)n− 2k−1
2
ex(n, Kk )
⌊5n/2 − 4⌋
2(k−1)n− 2k−1
2
ex(n, Kk )
Perles-Pinchasi,[BKV03]
[CP92]
Pach [PA95], Perles
ex(n, Kχc (G) )
ex(n, Kk )+n−k+1
(k − 1)n
⌊(k − 2)n/2⌋
ex(n,Kχc (G) )+o(n2 )
ex(n, Kk )+n−k+1
(k − 1)n
⌊(k − 2)n/2⌋
[BKV03]
[KP96]
[Kup84]
Perles [BKV03]
FIGURE 10.1.2
Convex geometric graph with n = 13 vertices and 6n− 72 = 57
edges, no 4 of which are pairwise crossing [CP92].
of this type can be generalized to complete geometric graphs, where the monochromatic subgraphs are required to satisfy certain geometric conditions.
1. Károlyi-Pach-Tóth theorem [KPT97]: If the edges of a finite complete geometric graph are colored by two colors, there exists a noncrossing spanning tree,
all of whose edges are of the same color. (This statement was conjectured by
Bialostocki and Dierker [BDV04]. The analogous assertion for abstract graph
follows from the fact that any graph or its complement is connected.)
2. Geometric Ramsey numbers: Let G1 , . . . , Gk be not necessarily different classes
of geometric graphs. Let R (G1 , . . . , Gk ) denote the smallest positive number
R with the property that any complete geometric graph of R vertices whose
edges are colored with k colors (1, . . . , k, say) contains, for some i, an i-colored
subgraph belonging to Gi . If G1 = . . . = Gk = G, we write R (G; k) instead of
R (G1 , . . . , Gk ). If k = 2, for the sake of simplicity, let R (G) stand for R (G; 2).
Some known results on the numbers R (G1 , G2 ) are listed in Table 10.1.3. In
line 3 of the table, we have a better result if we restrict our attention to convex
geometric graphs: For any 2-coloring of the edges of a complete convex geo-
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J. Pach
metric graph with 2k − 1 vertices, there exists a noncrossing monochromatic
path of length k ≥ 2, and this result cannot be improved. The upper bound
2(k − 1)(k − 2) + 2 in line 4 is tight for convex geometric graphs [BCK+ 15].
The bounds in line 4 also hold when G1 = G2 consists of all noncrossing cycles
of length k, triangulated from one of their vertices. The geometric Ramsey
numbers of convex geometric graphs, when G1 = G2 consists of all isomorphic copies of a given convex geometric graph with at most 4 vertices, can be
found in [BH96]. In [CGK+ 15], polynomial upper bounds are established for
the geometric Ramsey numbers of the “ladder graphs” Lk , consisting of two
paths of length k with an edge connecting each pair of corresponding vertices.
TABLE 10.1.3
G1
Geometric Ramsey numbers R (G1 , G2 ) from [KPT97] and [KPTV98].
all noncrossing trees
of k vertices
k disjoint edges
noncrossing paths
of length k
noncrossing cycles
of length k
G2
all noncrossing trees
of k vertices
l disjoint edges
noncrossing paths
of length k
noncrossing cycles
of length k
LOWER BOUND
UPPER BOUND
k
k
k + l + max{k, l} − 1
Ω(k)
k + l + max{k, l} − 1
O(k 3/2 )
(k − 1)2
2(k − 1)(k − 2) + 2
3. Pairwise disjoint copies: For any positive integer k, let kG denote the class of
all geometric graphs that can be obtained by taking the union of k pairwise
disjoint members of G. If k is a power of 2 then
R(kG) ≤ (R(G) + 1)k − 1.
In particular, if G = T is the class of triangles, we have R(T ) = 6. Thus, the
above bound yields that
R(kT ) ≤ 7k − 1,
provided that k is a power of 2. This result cannot be improved [KPTV98].
Furthermore, for any k > 0, we have
3(R(G) + 1)
R(G) + 1
R(kG) ≤
k−
.
2
2
For the corresponding quantities for convex geometric graphs, we have
Rc (kG) ≤ (Rc (G) + 1)k − 1.
4. Constructive vertex- and edge-Ramsey numbers: Given a class of geometric
graphs G, let Rv (G) denote the smallest number R such that there exists
a (complete) geometric graph of R vertices that, for any 2-coloring of its
edges, has a monochromatic subgraph belonging to G. Similarly, let Re (G)
denote the minimum number of edges of a geometric graph with this property.
Rv (G) and Re (G) are called the vertex- and edge-Ramsey number of G,
respectively. Clearly, we have
R (G)
Rv (K) ≤ R (G) , Re (G) ≤
.
2
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Chapter 10: Geometric graph theory
263
(For abstract graphs, similar notions are discussed in [EFRS78, Bec83].)
For Pk , the class of noncrossing paths of length k, we have Rv (Pk ) = O(k 3/2 )
and Re (Pk ) = O(k 2 ).
OPEN PROBLEMS
1. What is the smallest number u = u(n) such that there exists a “universal”
set U of u points in the plane with the property that every planar graph of
n vertices admits a noncrossing straight-line drawing on a suitable subset of
U [FPP90]? It follows from the existence of a small grid drawing (see above)
that u(n) ≤ n2 , and it is shown in [BCD+ 14] that u(n) ≤ n2 /4. From below
we have only u(n) > 1.235n; [Kur04]. Certain subclasses of planar graphs
admit universal sets of size o(n2 ); [FT15].
2. It was shown by Chalopin and Gonçalvez [CG09] that the vertices of every
planar graph G be represented by straight-line segments in the plane so that
two segments intersect if and only if the corresponding vertices are adjacent.
If the chromatic number of G is 2 or 3, then segments of 2 resp. 3 different
directions suffice [FMP94, FM07]. Does there exist a constant c such that the
vertices of every planar graph can be represented by segments using at most
c different directions?
3. (Erdős, Kaneko-Kano) What is the largest number A = A(n) such that any
set of n red and n blue points in the plane admits a noncrossing alternating
path of length A? It is known that A(n) ≤ (4/3 + o(1))n; [KPT07].
4. Is it true that, for any fixed k, the maximum number of edges of a geometric
graph with n vertices that does not have k pairwise crossing edges is O(n)?
5. (Aronov et al.) Is it true that any complete geometric graph with n vertices
has at least Ω(n) pairwise
crossing edges? It was shown in [AEG+ 94] that
p
one can always find n/12 pairwise crossing edges. On the other hand, any
complete geometric graph with n vertices has a noncrossing Hamiltonian path,
hence ⌊n/2⌋ pairwise disjoint edges.
6. (Larman-Matoušek-Pach-Törőcsik) What is the smallest positive number r =
r(n) such that any family of r closed segments in general position in the plane
has n members that are either pairwise disjoint or pairwise crossing? It is
known [LMPT94, Kyn12] that nlog 169/ log 8 ≈ n2.466 ≤ r(n) ≤ n5 .
10.2 CROSSING NUMBERS
The investigation of crossing numbers started during WWII with Turán’s Brick
Factory Problem [Tur77]: how should one redesign the routes of railroad tracks
between several kilns and storage places in a brick factory so as to minimize the
number of crossings? In the early eighties, it turned out that the chip area required
for the realization (VLSI layout) of an electrical circuit is closely related to the
crossing number of the underlying graph [Lei83]. This discovery gave an impetus
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J. Pach
to research in the subject. More recently, it has been realized that general bounds
on crossing numbers can be used to solve a large variety of problems in discrete
and computational geometry.
GLOSSARY
Drawing of a graph: A representation of the graph in the plane such that its
vertices are represented by distinct points and its edges by simple continuous
arcs connecting the corresponding point pairs. In a drawing (a) no edge passes
through any vertex other than its endpoints, (b) no two edges touch each other
(i.e., if two edges have a common interior point, then at this point they properly
cross each other), and (c) no three edges cross at the same point.
Crossing: A common interior point of two edges in a graph drawing. Two edges
may have several crossings.
Crossing number of a graph: The smallest number of crossings in any drawing
of G, denoted by cr(G). Clearly, cr(G) = 0 if and only if G is planar.
Rectilinear crossing number: The minimum number of crossings in a drawing
of G in which every edge is represented by a straight-line segment. It is denoted
by lin-cr(G).
Pairwise crossing number: The minimum number of crossing pairs of edges
over all drawings of G, denoted by pair-cr(G). (Here the edges can be represented by arbitrary continuous curves, so that two edges may cross more than
once, but every pair of edges can contribute at most one to pair-cr(G).)
Odd crossing number: The minimum number of those pairs of edges that cross
an odd number of times, over all drawings of G. It is denoted by odd-cr(G).
Biplanar crossing number: The minimum of cr(G1 ) + cr(G2 ) over all partitions of the graph into two edge-disjoint subgraphs G1 and G2 .
Bisection width: The minimum number b(G) of edges whose removal splits the
graph G into two roughly equal subgraphs. More precisely, b(G) is the minimum
number of edges running between V1 and V2 over all partitions of the vertex set
of G into two disjoint parts V1 ∪ V2 such that |V1 |, |V2 | ≥ |V (G)|/3.
Cut width: The minimum number c(G) such that there is a drawing of G in
which no two vertices have the same x-coordinate and every vertical line crosses
at most c(G) edges.
Path width: The minimum number p(G) such that there is a sequence of at
most (p(G) + 1)-element sets V1 , V2 , . . . , Vr ⊆ V (G) with the property that both
endpoints of every edge belong to some Vi and, if a vertex occurs in Vi and
Vk (i < k), then it also belongs to every Vj , i < j < k.
GENERAL ESTIMATES
Garey and Johnson [GJ83] showed that the determination of the crossing number
is an NP-complete problem. Analogous results hold for the rectilinear crossing
number [Bie91], for the pair crossing number [SSS̆03], and for the odd crossing
number [PT00b]. The exact determination of crossing numbers of relatively small
graphs of a simple structure (such as complete or complete bipartite graphs) is a
hopelessly difficult task, but there are several useful bounds. There is an algorithm
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Chapter 10: Geometric graph theory
265
[EGS03] for computing a drawing of a bounded-degree graph with n vertices, for
which n plus the the number of crossings is O(log3 n) times the optimum.
1. For a simple graph G with n ≥ 3 vertices and e edges, cr(G) ≥ e − 3n + 6.
From this inequality, a simple probabilistic argument shows that cr(G) ≥
ce3 /n2 , for a suitable positive constant c. This important bound, due to
Ajtai-Chvátal-Newborn-Szemerédi [ACNS82] and, independently, to Leighton
[Lei83], is often referred to as the crossing lemma. We know that 0.03 ≤ c ≤
0.09 [PT97, PRTT06, Ack16]. The lower bound follows from line 8 in Table
10.1.1. Similar statements hold for pair-cr(G) and odd-cr(G) [PT00b].
2. Crossing lemma for multigraphs [Szé97]: Let G be a multigraph with n
vertices and e edges, i.e., the same pair of vertices can be connected by more
than one edge. Let m denote the maximum multiplicity of an edge. Then
e3
− m2 n,
mn2
where c denotes the same constant as in the previous paragraph.
cr(G) ≥ c
3. Midrange crossing constant: Let κ(n, e) denote the minimum crossing number
of a graph G with n vertices and at least e edges. That is,
κ(n, e) =
min
cr(G).
n(G) = n
e(G) ≥ e
It follows from the crossing lemma that, for e ≥ 4n, κ(n, e)n2 /e3 is bounded
from below and from above by two positive constants. Erdős and Guy [EG73]
conjectured that if n ≪ e ≤ n2 /100, then lim κ(n, e)n2 /e3 exists. (We use
the notation f (n) ≫ g(n) to mean that limn→∞ f (n)/g(n) = ∞.) This was
partially settled in [PST00]: if n ≪ e ≪ n2 , then
lim κ(n, e)
n→∞
n2
=C>0
e3
exists. Moreover, the same result is true with the same constant C, for
drawings on every other orientable surface.
4. Graphs with monotone properties: A graph property P is said to be monotone if (i) for any graph G satisfying P, every subgraph of G also satisfies P;
and (ii) if G1 and G2 satisfy P, then their disjoint union also satisfies P. For
any monotone property P, let ex(n, P) denote the maximum number of edges
that a graph of n vertices can have if it satisfies P. In the special case when
P is the property that the graph does not contain a subgraph isomorphic to
a fixed forbidden subgraph H, we write ex(n, H) for ex(n, P).
Let P be a monotone graph property with ex(n, P) = O(n1+α ) for some
α > 0. In [PST00], it was proved that there exist two constants c, c′ > 0 such
that the crossing number of any graph G with property P that has n vertices
and e ≥ cn log2 n edges satisfies
cr(G) ≥ c′
Preliminary version (December 12, 2016).
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.
n1+1/α
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J. Pach
This bound is asymptotically tight, up to a constant factor. In particular, if
e > 4n and G has no cycle of length at most 2r, then the crossing number of
G satisfies
er+2
cr(G) ≥ cr r+1 ,
n
where cr > 0 is a suitable constant. For r = 2, 3, and 5, these bounds are
asymptotically tight, up to a constant factor. If G does not contain a complete
bipartite subgraph Kr,s with r and s vertices in its classes, s ≥ r, then we
have
e3+1/(r−1)
cr(G) ≥ cr,s 2+1/(r−1) ,
n
where cr,s > 0 is a suitable constant. These bounds are tight up to a constant
factor if r = 2, 3, or if r is arbitrary and s > (r − 1)!.
5. Crossing number vs. bisection width b(G): For any vertex v ∈ V (G), let d(v)
denote the degree of v in G. It was shown in [PSS96] and [SV94] that
cr(G) +
1 2
1 X 2
d (v) ≥
b (G).
16
40
v∈V (G)
A similar statement holds with a worse constant for the cut width c(G) of G
[DV02]. This, in turn, implies that the same is true for p(G), the path width
of G, as we have p(G) ≤ c(G) for every G [Kin92].
6. Relations between different crossing numbers: Clearly, we have
odd-cr(G) ≤ pair-cr(G) ≤ cr(G) ≤ lin-cr(G).
It was shown [BD93] that there are graphs with crossing number 4 whose
rectilinear crossing numbers are arbitrarily large.
It was established in [PT00b] and [Mat14] that
2
cr(G) ≤ 2 (odd-cr(G)) ,
cr(G) ≤ O (pair-cr(G))3/2 log2 odd-cr(G) ,
respectively. Pelsmajer et al. [PSS08] discovered a series of graphs for which
odd-cr(G) 6= pair-cr(G). In fact, there are graphs satisfying the inequality
pair-cr(G) ≥ 1.16odd-cr(G); see [Tót08]. We cannot rule out the possibility that
pair-cr(G) = cr(G)
for every graph G.
7. Crossing numbers of random graphs: Let G = G(n, p) be a random graph
with n vertices, whose edges are chosen independently with probability
p=
p(n). Let e denote the expected number of edges of G, i.e., e = p · n2 . It is
not hard to see that if e > 10n, then almost surely b(G) ≥ e/10. It therefore
follows from the above relation between the crossing number and the bisection
width that almost surely we have lin-cr(G) ≥ cr(G) ≥ e2 /4000. Evidently,
the order of magnitude of this bound cannot be improved. A similar inequality
was proved in [ST02] for the pairwise crossing number, under the stronger
condition that e > n1+ǫ for some ǫ > 0.
Preliminary version (December 12, 2016).
Chapter 10: Geometric graph theory
267
8. Biplanar crossing number vs. crossing number: It is known [CSSV08] that
the biplanar crossing number of every graph is at most 3/8 times its crossing
number. It is conjectured that the statement remains true with 7/24 in place
of 3/8.
9. Harary-Kainen-Schwenk conjecture [HKS73]: For every n ≥ m ≥ 3 and cycles
Cn and Cm , cr(Cn × Cm ) is equal to n(m − 2). This was proved in [GS04] for
every m and for all sufficiently large n. For the crossing number of the skeleton
of the n-dimensional hypercube Qn , we have 1/20 + o(1) ≤ cr(Qn )/4n ≤
163/1024 [FF00, SV93].
OPEN PROBLEMS
1. (Pach-Tóth) Is it true that pair-cr(G) = cr(G) for every graph G? Does
there exist a constant γ such that cr(G) ≤ γodd-cr(G)?
2. Albertson conjectured that the crossing number of every graph whose chromatic number is at least k is at least as large as the crossing number of
Kk . This conjecture was proved for k ≤ 18, and it is also known that
the crossing number of such a graph exceeds ck 4 , for a suitable constant
c. [ACF09, BT10, Ack16].
3. Zarankiewicz’s conjecture [Guy69]: The crossing number of the complete bipartite graph Kn,m with n and m vertices in its classes satisfies
jmk m − 1 jnk n − 1
cr(Kn,m ) =
·
·
·
.
2
2
2
2
Kleitman [Kle70] verified this conjecture in the special case when min{m, n} ≤
6 and Woodall [Woo93] for m = 7, n ≤ 10.
FIGURE 10.2.1
Complete bipartite graph K5,6 with 24 crossings.
It is also conjectured that the crossing number of the complete graph Kn
satisfies
1 jnk n − 1
n−2
n−3
·
·
·
.
cr(Kn ) =
4 2
2
2
2
This conjecture is known to be true if we restrict our attention to drawings
where every edge is represented by an x-monotone curve. [AAF+ 14, BFK15].
An old construction of Blažek and Koman [BK64] shows that equality can be
attained even for drawings of this type.
Preliminary version (December 12, 2016).
268
J. Pach
4. Rectilinear crossing numbers of complete graphs: Determine the value
κ = lim
n→∞
lin-cr(Kn )
.
n
4
The best known bounds 0.3799 < κ ≤ 0.3805 were found by Ábrego et
al. [ACF+ 10, ACF+ 12]; see also [LVW+ 04]. All known exact values of lin-cr(Kn )
are listed in Table 10.2.1. For n < 12, we have lin-cr(Kn ) = cr(Kn ), but
lin-cr(K12 ) > cr(K12 ) [ACF+ 12].
TABLE 10.2.1
Exact values of lin-cr(Kn ).
n
lin-cr(Kn )
n
lin-cr(Kn )
n
lin-cr(Kn )
4
5
6
7
8
9
10
11
12
0
1
3
9
19
36
62
102
153
13
14
15
16
17
18
19
20
21
229
324
447
603
798
1029
1318
1657
2055
22
23
24
25
26
27
30
2528
3077
3699
4430
5250
6180
9726
5. Let G = G(n, p) be a random graph with n vertices, whose
edges are chosen
independently with probability p = p(n). Let e = p · n2 . Is it true that the
pairwise crossing number, the odd crossing number, and the biplanar crossing
number are bounded from below by a constant times e2 , provided that e ≫ n?
10.3 GENERALIZATIONS
The concept of geometric graph can be generalized in two natural directions. Instead of straight-line drawings, we can consider curvilinear drawings. If we put
them at the focus of our investigations and we wish to emphasize that they are objects of independent interest rather than planar representations of abstract graphs,
we call these drawings topological graphs. In this sense, the results in the previous
section about crossing numbers belong to the theory of topological graphs. Instead
of systems of segments induced by a planar point set, we can also consider systems
of simplices in the plane or in higher-dimensional spaces. Such a system is called a
geometric hypergraph.
GLOSSARY
Topological graph: A graph drawn in the plane so that its vertices are distinct
points and its edges are simple continuous arcs connecting the corresponding
vertices. In a topological graph (a) no edge passes through any vertex other
than its endpoints, (b) any two edges have only a finite number of interior points
in common, at which they properly cross each other, and (c) no three edges cross
at the same point. (Same as drawing of a graph.)
Preliminary version (December 12, 2016).
Chapter 10: Geometric graph theory
269
Weakly isomorphic topological graphs: Two topological graphs, G and H,
such that there is an incidence-preserving one-to-one correspondence between
{V (G), E(G)} and {V (H), E(H)} in which two edges of G intersect if and only
if the corresponding edges of H do.
Thrackle: A topological graph in which any two nonadjacent edges cross precisely
once and no two adjacent edges cross.
Generalized thrackle: A topological graph in which any two nonadjacent edges
cross an odd number of times and any two adjacent edges cross an even number
of times (not counting their common endpoint).
d-Dimensional geometric r-hypergraph Hrd : A pair (V, E), where V is a set
of points in general position in d-space, and E is a set of closed (r−1)-dimensional simplices induced by some r-tuples of V . The sets V and E are called
the vertex set and (hyper)edge set of Hrd , respectively. Clearly, a geometric
graph is a 2-dimensional geometric 2-hypergraph.
Forbidden geometric hypergraphs: A class F of geometric hypergraphs not
permitted to be contained in the geometric hypergraphs under consideration.
Given a class F of forbidden geometric hypergraphs, exdr (F , n) denotes the maximum number of edges that a d-dimensional geometric r-hypergraph Hrd of n
vertices can have without containing a geometric subhypergraph belonging to F .
Nontrivial intersection: k simplices are said to have a nontrivial intersection
if their relative interiors have a point in common.
Crossing of k simplices: A common point of the relative interiors of k simplices,
all of whose vertices are distinct. The simplices are called crossing simplices
if such a point exists. A set of simplices may be pairwise crossing but not
necessarily crossing. If we want to emphasize that they all cross, we say that
they cross in the strong sense or, in brief, that they strongly cross.
TOPOLOGICAL GRAPHS
The fairly extensive literature on topological graphs focuses on very few special
questions, and there is no standard terminology. Most of the methods developed
for the study of geometric graphs break down for topological graphs, unless we make
some further structural assumptions. For example, many arguments go through for
x-monotone drawings such that any two edges cross at most once. Sometimes it is
sufficient to assume the latter condition.
1. An Erdős-Szekeres type theorem: A classical theorem of Erdős and Szekeres
states that every complete geometric graph with n vertices has a complete
geometric subgraph, weakly isomorphic to a convex complete graph Cm with
m ≥ c log n vertices. For complete topological graphs with n vertices, any two
of whose edges cross at most once, one can prove the existence of a complete
topological subgraph with m ≥ c log1/8 n vertices that is weakly isomorphic
either to a convex complete graph Cm or to a so-called twisted complete graph
Tm , as depicted in Figure 10.3.1 [PT03].
2. Every topological complete graph with n vertices, any two of whose edges
cross at most once, has a noncrossing subgraph isomorphic to any given tree
T with at most c log1/6 n vertices. In particular, it contains a noncrossing
path with at least c log1/6 n vertices [PT03].
Preliminary version (December 12, 2016).
270
J. Pach
FIGURE 10.3.1
The twisted drawing Tm discovered by Harborth and
Mengersen [HM92].
3. Number of topological complete graphs: Let Φ(n), Φ(n), and Φd (n) denote
the number of different (i.e., pairwise weakly nonisomorphic) geometric complete graphs, topological complete graphs, and topological complete graphs
in which every pair of edges cross at most d times, resp. We have log Φ(n) =
Θ(n log n), log Φ(n) = Θ(n4 ), Ω(n2 ) ≤ log Φ1 (n) ≤ O(n2 α(n)O(1) ) , where
α(n) denotes the (extremely slowly growing) inverse of the Ackermann function and Ω(n2 log n) ≤ log Φd (n) ≤ o(n4 ) for every d ≥ 2. [PT06, Kyn13].
4. Reducing the number of crossings [SS̆04]: Given an abstract graph G = (V, E)
and a set of pairs of edges P ⊆ E2 , we say that a topological graph K is a
weak realization of G if no pair of edges not belonging to P cross each other.
If G has a weak realization, then it also has a weak realization in which every
edge crosses at most 2|E| other edges. There is an almost matching lower
bound for this quantity [KM91].
5. Every cycle of length different from 4 can be drawn as a thrackle [Woo69].
A bipartite graph can be drawn in the plane as a generalized thrackle if and
only if it is planar [LPS97]. Every generalized thrackle with n > 2 vertices
has at most 2n − 2 edges, and this bound is sharp [CN00].
FIGURE 10.3.2
Cycles C5 and C10 drawn as thrackles.
GEOMETRIC HYPERGRAPHS
If we want to generalize the results in the first two sections to higher dimensional
geometric hypergraphs, we face some unexpected difficulties. Even if we restrict
our attention to systems of triangles induced by 3-dimensional point sets in general
position, it is not completely clear how a “crossing” should be defined. If two segments cross, they do not share an endpoint. Should this remain true for triangles?
In this subsection, we describe some scattered results in this direction, but it will
require further research to identify the key notions and problems.
1. Let Dkr denote the class of all geometric r-hypergraphs consisting of k pairwise disjoint edges (closed (r−1)-dimensional simplices). Let Ikr (respectively,
SI rk ) denote the class of all geometric r-hypergraphs consisting of k simplices,
any two of which have a nontrivial intersection (respectively, all of which are
Preliminary version (December 12, 2016).
Chapter 10: Geometric graph theory
271
strongly intersecting). Similarly, let Ckr (respectively, SC rk ) denote the class
of all geometric r-hypergraphs consisting of k pairwise crossing (respectively,
strongly crossing) edges. In Table 10.3.1, we summarize the known estimates
on exdr (F , n), the maximum number of hyperedges (or, simply, edges) that
a d-dimensional geometric r-hypergraph of n vertices can have without containing any forbidden subconfiguration belonging to F . We assume d ≥ 3.
In the first line of the table, the lower bound is conjectured to be tight. The
upper bounds in the second line are tight for d = 2, 3.
TABLE 10.3.1
r
d
d
d
d
d
d+1
d+1
d+1
Estimates on exdr (F , n), the maximum number of edges
of a d-dimensional geometric r-hypergraph of n vertices
containing no forbidden subconfigurations belonging to F .
F
Dkd
Ikd (k = 2, 3)
Ikd (k > 3)
C2d
Ckd (k > 2)
Ikd+1
SI d+1
k
C2d+1
LOWER BOUND
UPPER BOUND
d−1
Ω(nd−1 )
?
?
Ω(nd−1 )
nd−(1/k)
O(nd−1 )
O(nd−1 log n)
O(nd−1 )
?
O(nd−(1/d)
)
O(n⌈d/2⌉ )
O(n⌈d/2⌉ )
O(nd )
Ω(n⌈d/2⌉ )
Ω(n⌈d/2⌉ )
Ω(nd )
k−2
SOURCE
[AA89]
[DP98]
[Val98]
[DP98]
[DP98]
[BF87, DP98]
[BF87, DP98]
[DP98]
2. Akiyama-Alon theorem [AA89]: Let V = V1 ∪ . . . ∪ Vd (|V1 | = . . . = |Vd | = n)
be a dn-element set in general position in d-space, and let E consist of all
(d−1)-dimensional simplices having exactly one vertex in each Vi . Then E
contains n disjoint simplices. This result can be applied to deduce the upper
bound in the first line of Table 10.3.1.
r
d
3. Assume
δthat, for suitable constants
δ c1 and 0 ≤ δ ≤ 1, we have exr (SC k , n) <
n
n
c1 r /n and e ≥ (c1 + 1) r /n . Then there exists c2 > 0 such that the
minimum number of strongly crossing k-tuples of edges in a d-dimensional
r-hypergraph with n vertices and e edges is at least
γ
n γ n
e /
c2
,
kr
r
where γ = 1 + (k − 1)r/δ. This result can be used to deduce the upper bound
in line 5 of Table 10.3.1.
4. A Ramsey-type result [DP98]: Let us 2-color all (d−1)-dimensional simplices
induced by (d + 1)n − 1 points in general position in Rd . Then one can always
find n disjoint simplices of the same color. This result cannot be improved.
5. Convex geometric hypergraphs in the plane [Bra04]: If we choose triangles
from points in convex position in the plane, then the concept of isomorphism
is much clearer than in the higher-dimensional cases. Thus two triangles
without a common vertex can occur in three mutual positions, and we have
ex(n, ) = Θ(n3 ), ex(n, ) = Θ(n2 ), ex(n, ) = Θ(n2 ). Similarly, two
Preliminary version (December 12, 2016).
272
J. Pach
triangles with one common vertex can occur again in three positions and
we have ex(n, ) = Θ(n3 ), ex(n, ) = Θ(n2 ), ex(n, ) = Θ(n2 ), which
is surprising, since the underlying hypergraph has a linear Turán function.
Finally, two triangles with two common vertices have two possible positions,
and we have ex(n, ) = Θ(n3 ), ex(n, ) = Θ(n2 ). Larger sets of forbidden
subconvex geometric hypergraphs occur as the combinatorial core of several
combinatorial geometry problems.
OPEN PROBLEMS
1. (Ringel, Harborth) For any k, determine or estimate the smallest integer
n = n(k) for which there is a complete topological graph with n vertices,
every pair of whose edges intersect at most once (including possibly at their
common endpoints), and every edge of which crosses at least k others. It is
known that n(1) √
= 8, 7 ≤ n(2) ≤ 11, 7 ≤ n(3) ≤ 14, 7 ≤ n(4) ≤ 16, and
n(k) ≤ 4k/3 + O( k) [HT69]. Does n(k) = o(k) hold?
2. (Harborth) Is it true that in every complete topological graph with n vertices,
every pair of whose edges cross at most once (including possibly at their
common endpoints), there are at least 2n − 4 empty triangles [Har98]? (A
triangle bounded by all edges connecting three vertices is said to be empty, if
there is no point in its interior.) It is known that every complete topological
graph with the above property has at least n empty triangles [AHP+ 15],
[Rui15].
3. Conway conjectured that the number of edges of a thrackle cannot exceed
its number of vertices. It is known that every thrackle with n vertices has
at most 1.4n edges [Xu14]. Conway’s conjecture is true for thrackles drawn
by x-monotone edges [PS11], and for thrackles drawn drawn as outerplanar
graphs [CN08].
4. (Kalai) What is the maximum number µ(n) of hyperedges that a 3-dimensional geometric 3-hypergraph of n vertices can have, if any pair of its hyperedges
either are disjoint or share at most one vertex? Is it true that µ(n) = o(n2 )?
Károlyi and Solymosi [KS02] showed that µ(n) = Ω(n3/2 ).
10.4 SOURCES AND RELATED MATERIAL
SURVEYS
All results not given an explicit reference above may be traced in these surveys.
[PA95]: Monograph devoted to combinatorial geometry. Chapter 14 is dedicated
to geometric graphs.
[Pac99, Kár13]: Introduction to geometric graph theory and survey on Ramsey-type
problems in geometric graph theory, respectively.
Preliminary version (December 12, 2016).
Chapter 10: Geometric graph theory
273
[Pac91, DP98]: The first surveys of results in geometric graph theory and geometric
hypergraph theory, respectively.
[PT00a, Pac00, Szé04, SSSV97, Sch14]: Surveys on open problems and on crossing
numbers.
[BETT99]: Monograph on graph drawing algorithms.
[BMP05]: Survey of representative results and open problems in discrete geometry,
originally started by the Moser brothers.
[Grü72]: Monograph containing many results and conjectures on configurations and
arrangements of points and arcs.
RELATED CHAPTERS
Chapter
Chapter
Chapter
Chapter
Chapter
1:
5:
11:
28:
55:
Finite point configurations
Pseudoline arrangements
Euclidean Ramsey theory
Arrangements
Graph drawing
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